North Korea threatened on Monday to shoot down American warplanes even if they were not in the country’s airspace, stating that President Trump’s comments suggesting he would eradicate North Korea and its leaders were “a declaration of war.”

The warning, made by Foreign Minister Ri Yong-ho of North Korea in New York after a week of United Nations General Assembly meetings, escalated the invective-laced exchanges with Mr. Trump and appeared to further preclude the possibility of a diplomatic exit from the biggest foreign crisis the administration has faced.

Administration officials denied that the United States had declared war on the isolated, nuclear-armed country of 25 million people, with Sarah Huckabee Sanders, the White House press secretary, calling such a suggestion “absurd.”

She also said that “it’s never appropriate to shoot down another country’s aircraft when it’s over international waters” and that the United States wants a peaceful denuclearization of North Korea.

But Lt. Gen. H. R. McMaster, Mr. Trump’s national security adviser, said he saw no prospect of talks with North Korea that would allow its “de facto nuclear capability.” North Korea’s top leader, Kim Jong-un, has already ruled out giving up nuclear weapons.

“The whole world should clearly remember it was the U.S. who first declared war on our country,” Mr. Ri told reporters at a news conference outside his hotel as he was about to return home.

“Since the United States declared war on our country, we will have every right to make countermeasures, including the right to shoot down United States strategic bombers even when they are not inside the airspace border of our country,” he said.

North Korea last shot down an American warplane in 1969, during the Nixon administration, killing all 31 crew members of a spy plane that was flying off its coast.

Today, North Korea’s ability to make good on its threat is limited. Its air force is outdated, undertrained and frequently short of fuel. But the threat reinforced a fear that Pyongyang and Washington are hurtling toward a possible armed conflict, even an unintended one.

Mr. Ri’s reference to the declaration of war appeared to refer to Mr. Trump’s assertion in a Twitter message over the weekend that the North Korean leadership may not “be around much longer” if it continues its threats.

Mr. Ri said that the question of “who would be around much longer will be answered” by North Korea.

It is possible that North Korea’s foreign minister wanted to make clear that it, too, could threaten pre-emptive military action, just as the United States has repeatedly suggested in recent months.

But Mr. Trump’s tweet over the weekend appeared to go further, suggesting that mere threats, rather than a military attack, could drive him to wipe out the country. Whether that was one of his characteristic outbursts or a strategic effort to intimidate North Korea was not clear — even to some of his advisers.

“If the goal is to intimidate the North Koreans, it needs to be understood that they are really hard to intimidate,” said Evans J. R. Revere, a Korea expert who is a former deputy assistant secretary of state.

“They’re not used to an American president saying these things,” Mr. Revere said. “They’re also masters at responding when their leader is attacked.”

The escalation of threats came two days after American warplanes flew close to the North’s coast, going farther north of the Demilitarized Zone — the dividing line between North and South — than any other American air mission since the turn of the century. The Air Force advertised the exercise, which involved only American aircraft, as a direct response to North Korea’s accelerated missile launches and a nuclear test two weeks ago.

Mr. Ri, who is well connected to the country’s top leadership, also said last week that the North was considering conducting an atmospheric nuclear test in response to Mr. Trump’s threats, which would be the first by any nation in 37 years.

Video

The United States uses two different categories of missile defense to counter North Korea. Here’s how they work and — sometimes — how they don’t.

It is unclear whether the North is capable of such a test, which is far more complicated and dangerous than the underground testing it has done six times in the past 11 years. But a senior Trump administration official said over the weekend that the Pentagon and intelligence agencies were taking the threat seriously and devising possible responses — including pre-emptive military strikes — for the White House.

And Col. Robert Manning, a Pentagon spokesman, told reporters on Monday that if North Korea did not stop its provocative actions, “we will make sure that we provide options to the president to deal with North Korea.”

The 1950-53 Korean War ended in an armistice, not a peace treaty, so hostilities have merely been in abeyance. Since then the North has often said that the United States was bringing the two countries “to the brink of war.”

But Mr. Ri’s remark about downing American aircraft was new and raised the possibility of a clash, even if a North Korean attack failed. He also said that “all options will be on the operations table of the supreme leadership” of North Korea.

Political analysts said the Trump administration should consider Mr. Ri’s comments more than just verbal volleys.

“I think they’re dangerously close to some kind of a conflict with North Korea,” said Jae H. Ku, the director of the U.S. Korea Institute at the Johns Hopkins School of Advanced International Studies in Washington.

“This is something I feared,” he said. “When we go down this road, our escalation could lead to accidental shootouts, and it may not be so accidental.”

The increasing acrimony also alarmed China, North Korea’s biggest trading partner, which strongly opposes the North’s missile and nuclear tests but has repeatedly urged de-escalation. “We want things to calm down,” China’s United Nations ambassador, Liu Jieyi, was quoted by Reuters as saying on Monday.

Mr. Kim, in an unusually direct response, said last week that Mr. Trump’s insults to him and his country in his United Nations General Assembly speech amounted to the “most ferocious declaration of a war in history,” which warranted “the highest level of hard-line countermeasure in history.”

If North Korea were to shoot down a United States military aircraft, it would not be the first time since the Korean War. In April 1969, the North shot down an unarmed United States Lockheed EC-121 spy plane on a North Korean intelligence-gathering mission over the Sea of Japan. The 31 deaths amounted to one of the biggest single losses of American military lives during the Cold War.

North Korean state radio said at the time that the aircraft had penetrated “deep into the airspace” of the country. The Defense Department said the plane was 50 nautical miles off the North Korean coast.

In 1994, a United States Army helicopter was shot down, and one of its two pilots killed, after it strayed into North Korean airspace. The second pilot was freed after 13 days of captivity. North Korea claimed the helicopter had been on a spying mission, but the United States maintained that the intrusion was the result of a navigation error.

Hours after Mr. Ri spoke on Monday, General McMaster told a conference organized by the Institute for the Study of War in Washington that the administration had no set preconditions for talks with North Korea. But he said any new negotiations must be held “under conditions that are different from previous talks,” which he said were drawn out and yielded little progress.

General McMaster said the administration does not want to negotiate in a way that “locks in the status quo as the new normal.”

Asked about a maritime blockade as a military option, General McMaster said it would probably provoke retaliation by North Korea. He also said many people seem to think — incorrectly — that there is an easy military option.

“What we’re not assuming is that anything we do will go without some kind of response to the situation,” he said.

General McMaster would not comment on the North Korean threat to shoot down American warplanes.